Statistical inference for high dimensional regression via Constrained Lasso
This addresses the challenge of constructing confidence intervals in high-dimensional models for statisticians and data scientists, representing a novel method for a known bottleneck rather than an incremental improvement.
The paper tackles the problem of statistical inference for low-dimensional components in high-dimensional regression by proposing the Constrained Lasso (CLasso) estimator, which achieves asymptotically normal limits and attains the Cramér-Rao lower bound, with empirical studies showing encouraging performance.
In this paper, we propose a new method for estimation and constructing confidence intervals for low-dimensional components in a high-dimensional model. The proposed estimator, called Constrained Lasso (CLasso) estimator, is obtained by simultaneously solving two estimating equations---one imposing a zero-bias constraint for the low-dimensional parameter and the other forming an $\ell_1$-penalized procedure for the high-dimensional nuisance parameter. By carefully choosing the zero-bias constraint, the resulting estimator of the low dimensional parameter is shown to admit an asymptotically normal limit attaining the Cramér-Rao lower bound in a semiparametric sense. We propose a tuning-free iterative algorithm for implementing the CLasso. We show that when the algorithm is initialized at the Lasso estimator, the de-sparsified estimator proposed in van de Geer et al. [\emph{Ann. Statist.} {\bf 42} (2014) 1166--1202] is asymptotically equivalent to the first iterate of the algorithm. We analyse the asymptotic properties of the CLasso estimator and show the globally linear convergence of the algorithm. We also demonstrate encouraging empirical performance of the CLasso through numerical studies.